Theorem onexlimgt 43817

Description: For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)

Assertion

Ref	Expression
onexlimgt	⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem onexlimgt

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9601	. . . 4 ⊢ ω ∈ On
2		onun2 6456	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ∪ ω) ∈ On)
3	1, 2	mpan2 701	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∪ ω) ∈ On)
4		onexomgt 43815	. . 3 ⊢ ((𝐴 ∪ ω) ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
5	3, 4	syl 17	. 2 ⊢ (𝐴 ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
6		simp2 1150	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝑎 ∈ On)
7		omcl 8505	. . . . 5 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
8	1, 6, 7	sylancr 596	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (ω ·o 𝑎) ∈ On)
9		noel 4290	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∪ ω) ∈ ∅
10		oveq2 7404	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = (ω ·o ∅))
11		om0 8486	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → (ω ·o ∅) = ∅)
12	1, 11	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ω ·o ∅) = ∅
13	10, 12	eqtrdi 2813	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (ω ·o 𝑎) = ∅)
14	13	eleq2d 2848	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ (𝐴 ∪ ω) ∈ ∅))
15	14	notbid 320	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
16	15	adantl 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
17	9, 16	mpbiri 260	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
18	17	pm2.21d 121	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎)))
19	18	ex 416	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎))))
20	19	com23 86	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → (𝑎 = ∅ → Lim (ω ·o 𝑎))))
21	20	3impia 1130	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ → Lim (ω ·o 𝑎)))
22		limom 7862	. . . . . . . . 9 ⊢ Lim ω
23	1, 22	pm3.2i 474	. . . . . . . 8 ⊢ (ω ∈ On ∧ Lim ω)
24	6, 23	jctir 528	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)))
25		omlimcl2 43816	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
26	24, 25	sylan 589	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
27	26	ex 416	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (∅ ∈ 𝑎 → Lim (ω ·o 𝑎)))
28		on0eqel 6471	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
29	6, 28	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
30	21, 27, 29	mpjaod 871	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → Lim (ω ·o 𝑎))
31		simp1 1149	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ On)
32	31, 8	jca 519	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On))
33		simp3 1151	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
34		ssun1 4130	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ ω)
35	33, 34	jctil 527	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)))
36		ontr2 6394	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On) → ((𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎)))
37	32, 35, 36	sylc 65	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎))
38		limeq 6358	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (Lim 𝑥 ↔ Lim (ω ·o 𝑎)))
39		eleq2 2851	. . . . . 6 ⊢ (𝑥 = (ω ·o 𝑎) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (ω ·o 𝑎)))
40	38, 39	anbi12d 641	. . . . 5 ⊢ (𝑥 = (ω ·o 𝑎) → ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) ↔ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))))
41	40	rspcev 3581	. . . 4 ⊢ (((ω ·o 𝑎) ∈ On ∧ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
42	8, 30, 37, 41	syl12anc 847	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))
43	42	rexlimdv3a 3167	. 2 ⊢ (𝐴 ∈ On → (∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎) → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥)))
44	5, 43	mpd 15	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  Oncon0 6346  Lim wlim 6347  (class class class)co 7396  ωcom 7846   ·_o comu 8435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442

This theorem is referenced by: (None)